Rational Multiplication/Division

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Rational Multiplication/Division · Grade 7 · Worksheet 2

1. A rectangular swimming pool is drawn on a coordinate plane with vertices at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular shallow end is marked off with vertices at (0, 0), (8, 0), and (0, 12). What is the area of the deep end section of the pool? Answer: 192 Solution: Length = 20 units, Width = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has vertices at (0, 0), (8, 0), and (0, 12) Base = 8 units, Height = 12 units Area of triangle = (1/2) × base × height = (1/2) × 8 × 12 = 48 square units Deep end area = Total pool…
   Full step-by-step solution

   Step 1: Find the area of the entire rectangular pool Length = 20 units, Width = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units Step 2: Find the area of the triangular shallow end The triangle has vertices at (0, 0), (8, 0), and (0, 12) Base = 8 units, Height = 12 units Area of triangle = (1/2) × base × height = (1/2) × 8 × 12 = 48 square units Step 3: Find the area of the deep end section Deep end area = Total pool area - Shallow end area Deep end area = 240 - 48 = 192 square units The answer is 192.

2. (-3/4) × (8/9) ÷ (-2/3) = ? Answer: 1 Solution: (-3/4) × (8/9) ÷ (-2/3) Dividing by (-2/3) is the same as multiplying by its reciprocal. Reciprocal of (-2/3) is (-3/2).
   Full step-by-step solution

   Let's solve step by step. We start with: (-3/4) × (8/9) ÷ (-2/3) --- **Step 1: Understand the division** Dividing by (-2/3) is the same as multiplying by its reciprocal. Reciprocal of (-2/3) is (-3/2). So the expression becomes: (-3/4) × (8/9) × (-3/2) --- **Step 2: Multiply the first two fractions** (-3/4) × (8/9) = (-3 × 8) / (4 × 9) = (-24) / (36) Simplify (-24/36): Divide numerator and denominator by 12: (-24 ÷ 12) / (36 ÷ 12) = (-2) / 3 So after first multiplication: (-2/3) --- **Step 3: Multiply by the third fraction** (-2/3) × (-3/2) = [(-2) × (-3)] / [3 × 2] = (6) / (6) = 1 --- **Step 4: Final answer** The result is 1.

3. A construction company needs to calculate the total area of a rectangular plot of land that is 3/4 of a mile long and 2/5 of a mile wide. They also need to determine how many sections they can divide it into if each section must be exactly 1/10 of a square mile. What is the total area of the plot, and how many sections can they create? Answer: 3/10, 3 Solution: Find the total area of the rectangular plot. The area of a rectangle is length × width. Length = 3/4 mile Width = 2/5 mile Area = (3/4) × (2/5) Multiply the fractions.
   Full step-by-step solution

   Step 1: Find the total area of the rectangular plot. The area of a rectangle is length × width. Length = 3/4 mile Width = 2/5 mile Area = (3/4) × (2/5) Step 2: Multiply the fractions. Multiply the numerators: 3 × 2 = 6 Multiply the denominators: 4 × 5 = 20 So area = 6/20 square miles. Step 3: Simplify the fraction. Both 6 and 20 can be divided by 2: 6 ÷ 2 = 3 20 ÷ 2 = 10 So area = 3/10 square miles. Step 4: Determine how many sections of 1/10 square mile each can be made. Number of sections = Total area ÷ Area per section = (3/10) ÷ (1/10) Step 5: Dividing fractions: multiply by the reciprocal. (3/10) ÷ (1/10) = (3/10) × (10/1) Step 6: Multiply the fractions. Numerators: 3 × 10 = 30 Denominators: 10 × 1 = 10 So 30/10 = 3 sections. Final answer: Total area = 3/10 square miles Number of sections = 3

4. Charlotte is a marine biologist tracking the migration of a pod of whales. The pod swims at an average rate of -3.75 miles per hour (negative indicates moving south) for 4.8 hours each day. After 6 days, Charlotte calculates the total displacement of the pod from its starting point. What is the total displacement in miles? Answer: -108 Solution: Find the total hours the pod swims. 4.8 hours per day * 6 days = 28.8 hours total. Multiply the rate by the total time: -3.75 miles per hour * 28.8 hours.
   Full step-by-step solution

   Step 1: Find the total hours the pod swims. 4.8 hours per day * 6 days = 28.8 hours total. Step 2: Multiply the rate by the total time: -3.75 miles per hour * 28.8 hours. Step 3: Multiply the absolute values: 3.75 * 28.8 = 108. Step 4: Since one factor is negative, the product is negative: -108. The total displacement is -108 miles, meaning the pod is 108 miles south of the starting point. The answer is -108.

5. Noah is an engineer designing a new section of a subway tunnel. The tunnel boring machine digs at a rate of -7.5 meters per hour (the negative sign indicates it is moving downward into the ground). After 12.5 hours of continuous operation, the machine then reverses direction and moves upward at a rate of 4.2 meters per hour for 5.5 hours to adjust for a utility line. What is the final position of the tunnel boring machine relative to its starting point at ground level? Answer: -70.65 Solution: Calculate the downward distance. Downward rate = -7.5 meters per hour Time = 12.5 hours Downward distance = -7.5 × 12.5 = -93.75 meters Calculate the upward distance.
   Full step-by-step solution

   Step 1: Calculate the downward distance. Downward rate = -7.5 meters per hour Time = 12.5 hours Downward distance = -7.5 × 12.5 = -93.75 meters Step 2: Calculate the upward distance. Upward rate = 4.2 meters per hour Time = 5.5 hours Upward distance = 4.2 × 5.5 = 23.1 meters Step 3: Find the final position. Final position = Downward distance + Upward distance Final position = -93.75 + 23.1 = -70.65 meters The tunnel boring machine is at a depth of 70.65 meters below ground level, so the final position is -70.65 meters.

6. -2/3 × (5/4 ÷ -3/8) = ? Answer: 20/9 Solution: Start with the division inside the parentheses: 5/4 ÷ -3/8 Dividing by a fraction is the same as multiplying by its reciprocal: 5/4 × -8/3 Multiply the fractions: (5 × -8)/(4 × 3) = -40/12 Simplify the fraction: -40/12 = -10/3 Now multiply by -2/3: -2/3 × -10/3 Multiply the fractions: (-2 ×…
   Full step-by-step solution

   Step 1: Start with the division inside the parentheses: 5/4 ÷ -3/8 Step 2: Dividing by a fraction is the same as multiplying by its reciprocal: 5/4 × -8/3 Step 3: Multiply the fractions: (5 × -8)/(4 × 3) = -40/12 Step 4: Simplify the fraction: -40/12 = -10/3 Step 5: Now multiply by -2/3: -2/3 × -10/3 Step 6: Multiply the fractions: (-2 × -10)/(3 × 3) = 20/9 Step 7: The final answer is 20/9.

7. Liam is designing a rectangular garden for his school project. The garden's length is 3/4 of a kilometer and its width is 2/5 of a kilometer. He needs to calculate the total area to determine how much fertilizer to buy. What is the area of Liam's garden in square kilometers? Answer: 3/10 Solution: To find the area of a rectangle, we multiply the length by the width. Write down the given dimensions. Length = 3/4 km Width = 2/5 km Write the formula for area.
   Full step-by-step solution

   To find the area of a rectangle, we multiply the length by the width. Step 1: Write down the given dimensions. Length = 3/4 km Width = 2/5 km Step 2: Write the formula for area. Area = Length × Width Step 3: Substitute the given values into the formula. Area = (3/4) × (2/5) Step 4: Multiply the fractions. To multiply fractions, multiply the numerators together and multiply the denominators together. Numerator: 3 × 2 = 6 Denominator: 4 × 5 = 20 So, Area = 6/20 Step 5: Simplify the fraction. Find the greatest common factor (GCF) of 6 and 20. The GCF is 2. Divide both the numerator and the denominator by 2. 6 ÷ 2 = 3 20 ÷ 2 = 10 So, 6/20 simplifies to 3/10. Step 6: State the final answer. The area of Liam's garden is 3/10 square kilometers.